Existence of stationary fronts in a system of two coupled wave equations with spatial inhomogeneity.

We investigate the existence of stationary fronts in a coupled system of two sine-Gordon equations with a smooth, "hat-like" spatial inhomogeneity. The spatial inhomogeneity corresponds to a spatially dependent scaling of the sine-Gordon potential term. The inhomogeneous uncoupled sine-Gordon equation has stable stationary solutions that persist in the coupled system. Carrying out a numerical investigation it is found that stable fronts bifurcate from these inhomogeneous sine-Gordon fronts provided the coupling between the two inhomogeneous sine-Gordon equations is strong enough. In order to analytically study the emerging fronts, we first approximate the smooth spatial inhomogeneity by a piecewise constant function. With this approximation, we prove analytically the existence of a bifurcation point of the inhomogeneous sine-Gordon front and the emerging states. To complete the argument, we prove that transverse fronts for a piecewise constant inhomogeneity persist for the smooth "hat-like" spatial inhomogeneity using geometric singular perturbation theory.